Geoscience Reference
In-Depth Information
than volume in order to make most of the param-
eters positive.
The DLA parameters relate the variation of
the moduli to volume, or density, rather than to
temperature and pressure. This is useful since the
variations of density with temperature, pressure,
composition and phase are fairly well under-
stood. Furthermore, anharmonic properties tend
to be independent of temperature and pressure
at constant volume. The anharmonic parame-
ter known as the thermal Gruneisen parame-
ter
particularly for the bulk modulus, and variations
of seismic velocities are due primarily to changes
in the rigidity.
Intrinsic effects are more important for the
rigidity than for the bulk modulus. Geophys-
ical results on the radial and lateral varia-
tions of velocity and density provide constraints
on high-pressure--high-temperature equations of
state. Many of the thermodynamic properties
of the lower mantle, required for equation-of-
state modeling, can be determined directly from
the seismic data. The effect of pressure on the
coefficient of thermal expansion, the Gruneisen
parameters, the lattice conductivity and the tem-
perature derivatives of seismic-wave velocities
should be taken into account in the interpreta-
tion of seismic data and in convection and geoid
calculations.
The lateral variation of seismic velocity is
very large in the upper 200 km of the mantle
but decreases rapidly below this depth. Velocity
itself generally increases with depth below about
200 km. This suggests that temperature varia-
tions are more important in the shallow mantle
than at greater depth. Most of the mantle is above
the Debye temperature and therefore thermody-
namic properties may approach their classical
high-temperature limits. What is needed is pre-
cise data on variations of properties with temper-
ature at high pressure and theoretical treatments
of properties of solids at simultaneous high pres-
sure and temperature.
I use the following relations and notation:
γ
is
relatively
constant
from
material
to
material
as
well
as
relatively
independent
of
temperature.
Anelastic effects
Solids are not ideally elastic; the moduli depend
on frequency. This is known as 'dispersion' and
it introduces an additional temperature depen-
dency on the seismic-wave velocities. This can be
written
) Q max [2 E /
/
=
/
π
dln v
dln T
(1
2
RT ]
where v is a seismic velocity, T is absolute tem-
perature, Q max is the peak value of the absorp-
tion band and E is the activation energy. The
temperature dependence arises from the relax-
ation time, which is an activated parameter that
depends exponentially on temperature. Many
otherexamplesaregiveninthechapterondis-
sipation. The above example applies inside the
seismic absorption band.
K T ={
(
ln K T /∂
ln
ρ
) T =
(
K T /∂
P ) T =
K T } T
Seismic constraints on
thermodynamics of the lower mantle
( ln K S /∂ ln ρ ) T = ( K T / K S ) K S ={ K S } T
( ln K S /∂ ln ρ ) P =− ( α K S ) 1 ( K S /∂ T ) P
= δ S ={ K S } P
( ln G /∂ ln ρ ) T = ( K T / G )( G /∂ P ) T
= ( K T / G ) G = G ={ G } T
( ln G /∂ ln ρ ) P =− ( α G ) 1 ( G /∂ T ) P
= g ={ G } P
( ln α/∂ ln ρ ) T δ S + γ =−{ α } T
( ln K T /∂ T ) V = α [( ln K T /∂ ln ρ ) T
For most solids at normal conditions, the effect
of temperature on the elastic properties is
controlled mainly by the variation of volume.
Volume-dependent extrinsic effects dominate at
low pressure and high temperature. Under these
conditions one expects that the relative changes
in shear velocity, due to lateral temperature
gradients in the mantle, should be similar to
changes in compressional velocity. However, at
high pressure, this contribution is suppressed,
(
ln K T /∂
ln
ρ
) P ]
( K T δ T )
= α
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